Revision Linear Inequations Natural Numbers N Integers J Rational Q Irrational I Real Numbers R Complex C Algebraic and Graphical Solutions. By I Porter

Introduction An inequation is formed when two mathematical statements have an unequality sign between them.Common inequality signs: > is greater than ≥ is greater than or equal to < is less than ≤ is less than or equal to Inequations can have an infinite number of solutions. Solving inequations makes use of the following axioms of inequality for real numbers a, b and c. If a > b , then 1. a + c > b + c 5. ac < bc if c < 0 2. a - c > b - c 6. if c < 0 3. ac > bc if c > 0 4. if c > 0 Similar axioms also apply for a < b.

Solving Inequalities Inequations may be simplified by: 1. adding the same number to both sides. i.e. 10 > 3, then 10 + 2 > 3 + 2 2. subtracting the same number to both sides. i.e. 10 > 3, then 10 - 2 > 3 - 2 3. multiplying both sides by the same positive number. i.e. 10 > 3, then 10 x 2 > 3 x 2 i.e. 10 > 3, then 4. dividing both sides by the same positive number. In all cases above, the direction of the inequality remains the same. Also, the above statements apply when ‘>’ is replaced with ‘<‘. Special Cases The inequality sign must be reversed when: 1. multiplying both sides by the same negative number. i.e. 10 > 3, but 10 x -2 < 3 x -2 i.e. 10 > 3, then < 2. dividing both sides by the same negative number.

Graphical Solutions Algebra Graphical Number Line Solution x > 4 x 16 17 18 19 20 x -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 4 5 6 3 2 x < -5 x ≥ -10 x ≤ 18 Note: ‘>’ and ‘< ‘ use and open circle. Note: ‘≥’ and ‘≤‘ use and closed (dot) circle. You also only need to write in three (3) numbers to indicate location and order.

Examples Inequations are solve exactly the same way as equations, with two exception as stated by the axioms (5) and (6). [ reverse the inequality when x or  by a negative number ] Solve and graph a solution for the following: a) 4x - 5 < 23 Add 5 to both sides. b) 4(2 - x) ≥ 3x + 14 Expand Brackets. 4x < 28 Divide both sides by 4. 8 - 4x ≥ 3x + 14 Subtract 3x from both sides. x < 7 Open circle, arrow left. 8 - 7x ≥ 14 Subtract 8 from both sides. - 7x ≥ 6 Divide both sides by -7. 8 7 6 x Reverse inequality sign. Closed circle, arrow left. x Always move ALGEBRA to the LEFT SIDE

Examples: Solve and graph a solution for the following: b) Add 3 to both sides. Cross multiply denominators. Divide both sides by 2. Expand brackets. Subtract 5x from both sides. Open Circle Closed Circle Add 3 to both sides. Divide both sides by -2. Reverse inequality sign. Closed circle, arrow right. -2 -1 1 2 3 4 5 6 7 -3 -4 -5 x Always move ALGEBRA to the LEFT SIDE

Exercise: Solve and graph a solution for each the following: -4 -5 -6 x 4 3 2 x -2 -3 -4 x -5 -6 -7 x 6 7 8 5 4 x 5 -4 x Always move ALGEBRA to the LEFT SIDE